A multigrid-based preconditioned Krylov subspace method for the Helmholtz equation with PML

In this paper, we generalize the complex shifted Laplacian preconditioner to the complex shifted Laplacian-PML preconditioner for the Helmholtz equation with perfectly matched layer (Helmholtz-PML equation). The Helmholtz-PML equation is discretized by an optimal 9-point difference scheme, and the preconditioned linear system is solved by the Krylov subspace method, especially by the biconjugate gradient stabilized method (Bi-CGSTAB). The spectral analysis of the linear system is given, and a new matrix-based interpolation operator is proposed for the multigrid method, which is used to approximately invert the preconditioner. The numerical experiments are presented to illustrate the efficiency of the preconditioned Bi-CGSTAB method with the multigrid based on the new interpolation operator, also, numerical results are given for comparing the performance of the new interpolation operator with that of classic bilinear interpolation operator and the one suggested in Erlangga et al. (2006) [10].     

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time of the wave and one for its amplitude. The travel time is obtained by a solution of the factored eikonal equation, and the amplitude is obtained by solving a complex‐valued advection–diffusion–reaction equation. The reformulated equation is equivalent to the original Helmholtz equation, and the differences between the numerical solutions of these equations arise only from discretization errors. We develop an efficient multigrid solver for obtaining the amplitude given the travel time, which can be efficiently computed. This approach is advantageous because the amplitude is typically smooth in this case and, hence, more suitable for multigrid solvers than the standard Helmholtz discretization. We demonstrate that our second‐order advection–diffusion–reaction discretization is more accurate than the standard second‐order discretization at high wave numbers, as long as there are no reflections or caustics. Moreover, we show that using our approach, the problem can be solved more efficiently than using the common shifted Laplacian multigrid approach.     

Comparison between the shifted-Laplacian preconditioning and the controllability methods for computational acoustics

Processes that can be modelled with numerical calculations of acoustic pressure fields include medical and industrial ultrasound, echo sounding, and environmental noise. We present two methods for making these calculations based on Helmholtz equation. The first method is based directly on the complex-valued Helmholtz equation and an algebraic multigrid approximation of the discretized shifted-Laplacian operator; i.e. the damped Helmholtz operator as a preconditioner. The second approach returns to a transient wave equation, and finds the time-periodic solution using a controllability technique. We concentrate on acoustic problems, but our methods can be used for other types of Helmholtz problems as well. Numerical experiments show that the control method takes more CPU time, whereas the shifted-Laplacian method has larger memory requirement.     

A multigrid‐based shifted Laplacian preconditioner for a fourth‐order Helmholtz discretization

In this paper, an iterative solution method for a fourth‐order accurate discretization of the Helmholtz equation is presented. The method is a generalization of that presented in (SIAM J. Sci. Comput. 2006; 27 :1471–1492), where multigrid was employed as a preconditioner for a Krylov subspace iterative method. The multigrid preconditioner is based on the solution of a second Helmholtz operator with a complex‐valued shift. In particular, we compare preconditioners based on a point‐wise Jacobi smoother with those using an ILU(0) smoother, we compare using the prolongation operator developed by de Zeeuw in (J. Comput. Appl. Math. 1990; 33 :1–27) with interpolation operators based on algebraic multigrid principles, and we compare the performance of the Krylov subspace method Bi‐conjugate gradient stabilized with the recently introduced induced dimension reduction method, IDR(s). These three improvements are combined to yield an efficient solver for heterogeneous problems. Copyright © 2009 John Wiley & Sons, Ltd.     

高波数波动问题的多水平方法

本文主要讨论求解高波数Helmholtz方程的多水平方法,主要回顾了一些具有代表性的多重网格方法.如Erlangga等人的shifted Laplacian预处理的多重网格法;Elman等提出的修正的多重网格方法;以及我们的基于连续内罚有限元(CIP-FEM)离散代数系统的多水平算法.最后还介绍了求解高波数时谐Maxwell方程的CIP-FEM离散代数系统的多水平算法.     

Optimal Multigrid Methods with New Transfer Operators Based on Finite Difference Approximations

We constructed new interpolation operator in multigrid methods, which is efficient to transfer residual error from coarse grid to fine grid. This operator used idea of solving local residual equation using the standard stencil and the skewed stencil of the centered difference approximation to the Laplacian operator. We also compared our new multigrid methods with traditional multigrid methods, and found that new method is optimal.     

Integral representations in quaternionic analysis related to the helmholtz operator

As the Laplacian also the Helmholtz operator can be factorized in Clifford analysis. Using the fundamental solution of the Helmholtz equation, a fundamental solution can be constructed for the factors of the Helmholtz operator. These are used in the case of quaternions to prove Cauchy–Pompeiu type representation formulas in terms of powers of the factors as well as in terms of powers of the Helmholtz operator.     

A new level‐dependent coarse grid correction scheme for indefinite Helmholtz problems

In this paper, we construct and analyze a level‐dependent coarse grid correction scheme for indefinite Helmholtz problems. This adapted multigrid (MG) method is capable of solving the Helmholtz equation on the finest grid using a series of MG cycles with a grid‐dependent complex shift, leading to a stable correction scheme on all levels. It is rigorously shown that the adaptation of the complex shift throughout the MG cycle maintains the functionality of the two‐grid correction scheme, as no smooth modes are amplified in or added to the error. In addition, a sufficiently smoothing relaxation scheme should be applied to ensure damping of the oscillatory error components. Numerical experiments on various benchmark problems show the method to be competitive with or even outperform the current state‐of‐the‐art MG‐preconditioned Krylov methods, for example, complex shifted Laplacian preconditioned flexible GMRES. Copyright © 2013 John Wiley & Sons, Ltd.     

A Vanka-based parameter-robust multigrid relaxation for the Stokes–Darcy Brinkman problems

We consider a block-structured multigrid method based on Braess–Sarazin relaxation for solving the Stokes–Darcy Brinkman equations discretized by the marker and cell scheme. In the relaxation scheme, an element-based additive Vanka operator is used to approximate the inverse of the corresponding shifted Laplacian operator involved in the discrete Stokes–Darcy Brinkman system. Using local Fourier analysis, we present the stencil for the additive Vanka smoother and derive an optimal smoothing factor for Vanka-based Braess–Sarazin relaxation for the Stokes–Darcy Brinkman equations. Although the optimal damping parameter is dependent on meshsize and physical parameter, it is very close to one. In practice, we find that using three sweeps of Jacobi relaxation on the Schur complement system is sufficient. Numerical results of two-grid and V(1,1)-cycle are presented, which show high efficiency of the proposed relaxation scheme and its robustness to physical parameters and the meshsize. Using a damping parameter equal to one gives almost the same convergence results as these for the optimal damping parameter.     

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q₂‐Q₁ (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.     

Operator splitting methods for pricing American options under stochastic volatility

We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.     

Solving time‐periodic fractional diffusion equations via diagonalization technique and multigrid

This paper addresses numerical computation of time‐periodic diffusion equations with fractional Laplacian. Time‐periodic differential equations present fundamental challenges for numerical computation because we have to consider all the discrete solutions once in all instead of one by one. An idea based on the diagonalization technique is proposed, which yields a direct parallel‐in‐time computation for all the discrete solutions. The major computation cost is therefore reduced to solve a series of independent linear algebraic systems with complex coefficients, for which we apply a multigrid method using the damped Richardson iteration as the smoother. Such a linear solver possesses mesh‐independent convergence factor, and we make an optimization for the damping parameter to minimize such a constant convergence factor. Numerical results are provided to support our theoretical analysis.     

Multigrid Methods with Newton-Gauss-Seidel Smoothing and Constraint Preserving Interpolation for Obstacle Problems

In this paper, we propose a multigrid algorithm based on the full approximate scheme for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A Newton-Gauss-Seidel (NGS) method is used as smoother. A Galerkin coarse grid operator is proposed for the membrane constrained obstacle problem. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. A special prolongation operator is used to interpolate functions accurately from the coarse grid to the fine grid at the boundary between the active and inactive sets. We will demonstrate the fast convergence of the proposed multigrid method for solving two model obstacle problems and compare the results with other multigrid methods.     

Multi‐periodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n‐torus

In this paper, we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the n‐torus. Using the Clifford algebra calculus, the solutions can be expressed as multi‐periodic eigensolutions to the Dirac operator associated with a complex parameter λ∈?. Physically, these can be interpreted as the solutions to the time‐harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular, we set up Green‐type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application, we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half cylinder. Finally, we introduce hypercomplex integral operators on these manifolds, which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the n‐torus. Copyright © 2009 John Wiley & Sons, Ltd.     

A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.     

Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface,I: One Dimensional Problems

In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface. Second order accuracy for the first derivative is obtained as well. The method is based on the Ghost Fluid Method, making use of ghost points on which the value is defined by suitable interface conditions. The multi-domain formulation is adopted, where the problem is split in two sub-problems and interface conditions will be enforced to close the problem. Interface conditions are relaxed together with the internal equations (following the approach proposed in [10] in the case of smooth coefficients), leading to an iterative method on all the set of grid values (inside points and ghost points). A multigrid approach with a suitable definition of the restriction operator is provided. The restriction of the defect is performed separately for both sub-problems, providing a convergence factor close to the one measured in the case of smooth coefficient and independent on the magnitude of the jump in the coefficient. Numerical tests will confirm the second order accuracy. Although the method is proposed in one dimension, the extension in higher dimension is currently underway [12] and it will be carried out by combining the discretization of [10] with the multigrid approach of [11] for Elliptic problems with non-eliminated boundary conditions in arbitrary domain.     

Multigrid for second‐order ADN elliptic systems

This paper theoretically examines a multigrid strategy for solving systems of elliptic partial differential equations (PDEs) introduced in the work of Lee. Unlike most multigrid solvers that are constructed directly from the whole system operator, this strategy builds the solver using a factorization of the system operator. This factorization is composed of an algebraic coupling term and a diagonal (decoupled) differential operator. Exploiting the factorization, this approach can produce decoupled systems on the coarse levels. The corresponding coarse‐grid operators are in fact the Galerkin variational coarsening of the diagonal differential operator. Thus, rather than performing delicate coarse‐grid selection and interpolation weight procedures on the original strongly coupled system as often done, these procedures are isolated to the diagonal differential operator. To establish the theoretical results, however, we assume that these systems of PDEs are elliptic in the Agmon–Douglis–Nirenberg (ADN) sense and apply the factorization and multigrid only to the principal part of the system of PDEs. Two‐grid error bounds are established for the iteration applied to the complete system of PDEs. Numerical results are presented to illustrate the effectiveness of this strategy and to expose factors that affect the convergence of the methods derived from this strategy.     

Comparison of multigrid algorithms for high‐order continuous finite element discretizations

We present a comparison of different multigrid approaches for the solution of systems arising from high‐order continuous finite element discretizations of elliptic partial differential equations on complex geometries. We consider the pointwise Jacobi, the Chebyshev‐accelerated Jacobi, and the symmetric successive over‐relaxation smoothers, as well as elementwise block Jacobi smoothing. Three approaches for the multigrid hierarchy are compared: (1) high‐order h‐multigrid, which uses high‐order interpolation and restriction between geometrically coarsened meshes; (2) p‐multigrid, in which the polynomial order is reduced while the mesh remains unchanged, and the interpolation and restriction incorporate the different‐order basis functions; and (3) a first‐order approximation multigrid preconditioner constructed using the nodes of the high‐order discretization. This latter approach is often combined with algebraic multigrid for the low‐order operator and is attractive for high‐order discretizations on unstructured meshes, where geometric coarsening is difficult. Based on a simple performance model, we compare the computational cost of the different approaches. Using scalar test problems in two and three dimensions with constant and varying coefficients, we compare the performance of the different multigrid approaches for polynomial orders up to 16. Overall, both h‐multigrid and p‐multigrid work well; the first‐order approximation is less efficient. For constant coefficients, all smoothers work well. For variable coefficients, Chebyshev and symmetric successive over‐relaxation smoothing outperform Jacobi smoothing. While all of the tested methods converge in a mesh‐independent number of iterations, none of them behaves completely independent of the polynomial order. When multigrid is used as a preconditioner in a Krylov method, the iteration number decreases significantly compared with using multigrid as a solver. Copyright © 2015 John Wiley & Sons, Ltd.     

H least-squares method for the velocity–pressure–stress formulation of Stokes equations

This paper studies the discrete H⁻¹-norm least-squares method for the incompressible Stokes equations based on the velocity–pressure–stress formulation by the least-squares functional defined as the sum of L²-norms and H⁻¹-norm of the residual equations. Some computational experiments by multigrid method and preconditioning conjugate gradient method (PCGM) on this method are shown by taking efficient and β in the discrete solution operator T_h=h²I+βB_h corresponding to the minus one norm. We also propose a new method and compare it with PCGM and multigrid method through the analysis of numerical experiments depending on the choice of β.     

A Black-Box Multigrid Preconditioner for the Biharmonic Equation

We examine the convergence characteristics of a preconditioned Krylov subspace solver applied to the linear systems arising from low-order mixed finite element approximation of the biharmonic problem. The key feature of our approach is that the preconditioning can be realized using any “black-box” multigrid solver designed for the discrete Dirichlet Laplacian operator. This leads to preconditioned systems having an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. Numerical results show that the performance of the methodology is competitive with that of specialized fast iteration methods that have been developed in the context of biharmonic problems. This revised version was published online in July 2006 with corrections to the Cover Date.